To solve for the value of the variable [tex]\( x \)[/tex] in simplest radical form, we need to understand the context of the problem. The problem likely refers to a geometric context involving a 30-60-90 triangle, given the angles 60° and the relationship involving [tex]\( \sqrt{3} \)[/tex].
### Step-by-Step Solution:
1. 30-60-90 Triangle Properties:
A 30-60-90 triangle is a special right triangle. The sides are in a fixed ratio:
- The side opposite the 30° angle (short leg) is [tex]\( a \)[/tex].
- The side opposite the 60° angle (long leg) is [tex]\( a\sqrt{3} \)[/tex].
- The hypotenuse is [tex]\( 2a \)[/tex].
2. Identifying the sides:
Suppose the problem involves a 30-60-90 triangle where the given measures can be mapped to this special ratio.
3. Analyzing the Values Provided:
Based on the given information:
- [tex]\( x = 20 \sqrt{\sqrt{3}} \)[/tex] is provided.
- Given the angles 60° and possibly hinting at the sides linked to them, we translate these into the triangle's side properties.
4. Simplification of [tex]\( x \)[/tex]:
[tex]\( x = 20 \sqrt{\sqrt{3}} \)[/tex] seems to be an unexpected form; usually, we have [tex]\( \sqrt{3} \)[/tex]. But it hints a further simplification is necessary:
[tex]\[
x = 20 \sqrt{\sqrt{3}}
\][/tex]
Notice the nested square root. Simplify by considering how that resolves:
[tex]\[
\sqrt{\sqrt{3}} = (\sqrt{3})^{\frac{1}{2}} = 3^{\frac{1}{4}}
\][/tex]
Therefore:
[tex]\[
x = 20 \cdot 3^{\frac{1}{4}}
\][/tex]
5. Final Rationalized Form:
For exact values in simplest radical form:
[tex]\[
x = 20 \sqrt[4]{3}
\][/tex]
Here, [tex]\( \sqrt[4]{3} \)[/tex] is the fourth root of 3.
### Summary:
The value of the variable [tex]\( x \)[/tex] in simplest radical form, with the denominator rationalized for clarity, is:
[tex]\[ x = 20 \sqrt[4]{3} \][/tex]
This rational form expresses the value exactly where the nested radicals simplify to their basic radical terms.
